The quasi-Hadamard products of certain subclasses of analytic \(p\)-valent functions with negative coefficients
Let \(F_p(n,\lambda,\alpha)\), \(0\leq\alpha<p\), \(\lambda\geq0\), denote the class of functions \(f(z)=z^p-\sum_{k=p+n}^{\infty}a_kz^k\), \(a_k\geq0\), \(| z| <1\), which are \(p\)-valent and satisfy \[ \text{Re}\left((1-\lambda)\frac{f(z)}{z^p}+ \lambda\frac{f'(z)}{pz^{p-1}}\right)>\frac{\alpha}{p}. \] Similarly, the class \(G_p(n,\lambda,\alpha)\) is defined by \[ \text{Re}\left((p+\lambda(1-p))\frac{f'(z)}{pz^{p-1}}+ \lambda\frac{f"(z)}{pz^{p-2}}\right)>0. \] For \(f_j(z)=z^p-\sum_{k=p+n}^{\infty}a_{kj}z^k\), \(a_{kj}\geq0\), let \((f_1*f_2)(z)\) denote the quasi-Hadamard product \(z^p-\sum_{k=p+n}^{\infty}a_{k1}a_{k2}z^k\). Theorem 1 sharply states that if \(f_j(z)\in F_p(n,\lambda,\alpha_j)\), \(j=1,\dots,m\), then \((f_1*\dots*f_m)(z)\in F_p(n,\lambda,\gamma)\) where \[ \gamma=p-\frac{\prod_{j=1}^m(p-\alpha_j)}{(p+\lambda n)^{m-1}}. \] Theorem 2 and Theorem 4 formulate the corresponding results for \(f_j(z)\in F_p(n,\lambda,\alpha_j)\) or \(f_j(z)\in G_p(n,\lambda,\alpha_j)\) so that \((f_1*\dots*f_m)(z)\in G_p(n,\lambda,\gamma)\) for certain \(\gamma\).
scientific article; zbMATH DE number 5163906